Point Group C28h



C28h E C28 C14 (C28)3 C7 (C28)5 (C14)3 C4 (C7)2 (C28)9 (C14)5 (C28)11 (C7)3 (C28)13 C2 (C28)15 (C7)4 (C28)17 (C14)9 (C28)19 (C7)5 (C4)3 (C14)11 (C28)23 (C7)6 (C28)25 (C14)13 (C28)27 i (S28)15 (S7)11 (S28)17 (S14)9 (S28)19 (S7)5 (S4)3 (S14)11 (S28)23 (S7)13 (S28)25 (S14)13 (S28)27 σh S28 S14 (S28)3 S7 (S28)5 (S14)3 S4 (S7)9 (S28)9 (S14)5 (S28)11 (S7)3 (S28)13
Ag 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Bg 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
E1g* 2 1.9499 1.8019 1.5637 1.2470 0.8678 0.4450 0 -0.4450 -0.8678 -1.2470 -1.5637 -1.8019 -1.9499 -2 -1.9499 -1.8019 -1.5637 -1.2470 -0.8678 -0.4450 0 0.4450 0.8678 1.2470 1.5637 1.8019 1.9499 2 1.9499 1.8019 1.5637 1.2470 0.8678 0.4450 0 -0.4450 -0.8678 -1.2470 -1.5637 -1.8019 -1.9499 -2 -1.9499 -1.8019 -1.5637 -1.2470 -0.8678 -0.4450 0 0.4450 0.8678 1.2470 1.5637 1.8019 1.9499
E2g* 2 1.8019 1.2470 0.4450 -0.4450 -1.2470 -1.8019 -2 -1.8019 -1.2470 -0.4450 0.4450 1.2470 1.8019 2 1.8019 1.2470 0.4450 -0.4450 -1.2470 -1.8019 -2 -1.8019 -1.2470 -0.4450 0.4450 1.2470 1.8019 2 1.8019 1.2470 0.4450 -0.4450 -1.2470 -1.8019 -2 -1.8019 -1.2470 -0.4450 0.4450 1.2470 1.8019 2 1.8019 1.2470 0.4450 -0.4450 -1.2470 -1.8019 -2 -1.8019 -1.2470 -0.4450 0.4450 1.2470 1.8019
E3g* 2 1.5637 0.4450 -0.8678 -1.8019 -1.9499 -1.2470 0 1.2470 1.9499 1.8019 0.8678 -0.4450 -1.5637 -2 -1.5637 -0.4450 0.8678 1.8019 1.9499 1.2470 0 -1.2470 -1.9499 -1.8019 -0.8678 0.4450 1.5637 2 1.5637 0.4450 -0.8678 -1.8019 -1.9499 -1.2470 0 1.2470 1.9499 1.8019 0.8678 -0.4450 -1.5637 -2 -1.5637 -0.4450 0.8678 1.8019 1.9499 1.2470 0 -1.2470 -1.9499 -1.8019 -0.8678 0.4450 1.5637
E4g* 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470
E5g* 2 0.8678 -1.2470 -1.9499 -0.4450 1.5637 1.8019 0 -1.8019 -1.5637 0.4450 1.9499 1.2470 -0.8678 -2 -0.8678 1.2470 1.9499 0.4450 -1.5637 -1.8019 0 1.8019 1.5637 -0.4450 -1.9499 -1.2470 0.8678 2 0.8678 -1.2470 -1.9499 -0.4450 1.5637 1.8019 0 -1.8019 -1.5637 0.4450 1.9499 1.2470 -0.8678 -2 -0.8678 1.2470 1.9499 0.4450 -1.5637 -1.8019 0 1.8019 1.5637 -0.4450 -1.9499 -1.2470 0.8678
E6g* 2 0.4450 -1.8019 -1.2470 1.2470 1.8019 -0.4450 -2 -0.4450 1.8019 1.2470 -1.2470 -1.8019 0.4450 2 0.4450 -1.8019 -1.2470 1.2470 1.8019 -0.4450 -2 -0.4450 1.8019 1.2470 -1.2470 -1.8019 0.4450 2 0.4450 -1.8019 -1.2470 1.2470 1.8019 -0.4450 -2 -0.4450 1.8019 1.2470 -1.2470 -1.8019 0.4450 2 0.4450 -1.8019 -1.2470 1.2470 1.8019 -0.4450 -2 -0.4450 1.8019 1.2470 -1.2470 -1.8019 0.4450
E7g* 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0
E8g* 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450
E9g* 2 -0.8678 -1.2470 1.9499 -0.4450 -1.5637 1.8019 0 -1.8019 1.5637 0.4450 -1.9499 1.2470 0.8678 -2 0.8678 1.2470 -1.9499 0.4450 1.5637 -1.8019 0 1.8019 -1.5637 -0.4450 1.9499 -1.2470 -0.8678 2 -0.8678 -1.2470 1.9499 -0.4450 -1.5637 1.8019 0 -1.8019 1.5637 0.4450 -1.9499 1.2470 0.8678 -2 0.8678 1.2470 -1.9499 0.4450 1.5637 -1.8019 0 1.8019 -1.5637 -0.4450 1.9499 -1.2470 -0.8678
E10g* 2 -1.2470 -0.4450 1.8019 -1.8019 0.4450 1.2470 -2 1.2470 0.4450 -1.8019 1.8019 -0.4450 -1.2470 2 -1.2470 -0.4450 1.8019 -1.8019 0.4450 1.2470 -2 1.2470 0.4450 -1.8019 1.8019 -0.4450 -1.2470 2 -1.2470 -0.4450 1.8019 -1.8019 0.4450 1.2470 -2 1.2470 0.4450 -1.8019 1.8019 -0.4450 -1.2470 2 -1.2470 -0.4450 1.8019 -1.8019 0.4450 1.2470 -2 1.2470 0.4450 -1.8019 1.8019 -0.4450 -1.2470
E11g* 2 -1.5637 0.4450 0.8678 -1.8019 1.9499 -1.2470 0 1.2470 -1.9499 1.8019 -0.8678 -0.4450 1.5637 -2 1.5637 -0.4450 -0.8678 1.8019 -1.9499 1.2470 0 -1.2470 1.9499 -1.8019 0.8678 0.4450 -1.5637 2 -1.5637 0.4450 0.8678 -1.8019 1.9499 -1.2470 0 1.2470 -1.9499 1.8019 -0.8678 -0.4450 1.5637 -2 1.5637 -0.4450 -0.8678 1.8019 -1.9499 1.2470 0 -1.2470 1.9499 -1.8019 0.8678 0.4450 -1.5637
E12g* 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019
E13g* 2 -1.9499 1.8019 -1.5637 1.2470 -0.8678 0.4450 0 -0.4450 0.8678 -1.2470 1.5637 -1.8019 1.9499 -2 1.9499 -1.8019 1.5637 -1.2470 0.8678 -0.4450 0 0.4450 -0.8678 1.2470 -1.5637 1.8019 -1.9499 2 -1.9499 1.8019 -1.5637 1.2470 -0.8678 0.4450 0 -0.4450 0.8678 -1.2470 1.5637 -1.8019 1.9499 -2 1.9499 -1.8019 1.5637 -1.2470 0.8678 -0.4450 0 0.4450 -0.8678 1.2470 -1.5637 1.8019 -1.9499
Au 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
Bu 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1
E1u* 2 1.9499 1.8019 1.5637 1.2470 0.8678 0.4450 0 -0.4450 -0.8678 -1.2470 -1.5637 -1.8019 -1.9499 -2 -1.9499 -1.8019 -1.5637 -1.2470 -0.8678 -0.4450 0 0.4450 0.8678 1.2470 1.5637 1.8019 1.9499 -2 -1.9499 -1.8019 -1.5637 -1.2470 -0.8678 -0.4450 0 0.4450 0.8678 1.2470 1.5637 1.8019 1.9499 2 1.9499 1.8019 1.5637 1.2470 0.8678 0.4450 0 -0.4450 -0.8678 -1.2470 -1.5637 -1.8019 -1.9499
E2u* 2 1.8019 1.2470 0.4450 -0.4450 -1.2470 -1.8019 -2 -1.8019 -1.2470 -0.4450 0.4450 1.2470 1.8019 2 1.8019 1.2470 0.4450 -0.4450 -1.2470 -1.8019 -2 -1.8019 -1.2470 -0.4450 0.4450 1.2470 1.8019 -2 -1.8019 -1.2470 -0.4450 0.4450 1.2470 1.8019 2 1.8019 1.2470 0.4450 -0.4450 -1.2470 -1.8019 -2 -1.8019 -1.2470 -0.4450 0.4450 1.2470 1.8019 2 1.8019 1.2470 0.4450 -0.4450 -1.2470 -1.8019
E3u* 2 1.5637 0.4450 -0.8678 -1.8019 -1.9499 -1.2470 0 1.2470 1.9499 1.8019 0.8678 -0.4450 -1.5637 -2 -1.5637 -0.4450 0.8678 1.8019 1.9499 1.2470 0 -1.2470 -1.9499 -1.8019 -0.8678 0.4450 1.5637 -2 -1.5637 -0.4450 0.8678 1.8019 1.9499 1.2470 0 -1.2470 -1.9499 -1.8019 -0.8678 0.4450 1.5637 2 1.5637 0.4450 -0.8678 -1.8019 -1.9499 -1.2470 0 1.2470 1.9499 1.8019 0.8678 -0.4450 -1.5637
E4u* 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 -2 -1.2470 0.4450 1.8019 1.8019 0.4450 -1.2470 -2 -1.2470 0.4450 1.8019 1.8019 0.4450 -1.2470 -2 -1.2470 0.4450 1.8019 1.8019 0.4450 -1.2470 -2 -1.2470 0.4450 1.8019 1.8019 0.4450 -1.2470
E5u* 2 0.8678 -1.2470 -1.9499 -0.4450 1.5637 1.8019 0 -1.8019 -1.5637 0.4450 1.9499 1.2470 -0.8678 -2 -0.8678 1.2470 1.9499 0.4450 -1.5637 -1.8019 0 1.8019 1.5637 -0.4450 -1.9499 -1.2470 0.8678 -2 -0.8678 1.2470 1.9499 0.4450 -1.5637 -1.8019 0 1.8019 1.5637 -0.4450 -1.9499 -1.2470 0.8678 2 0.8678 -1.2470 -1.9499 -0.4450 1.5637 1.8019 0 -1.8019 -1.5637 0.4450 1.9499 1.2470 -0.8678
E6u* 2 0.4450 -1.8019 -1.2470 1.2470 1.8019 -0.4450 -2 -0.4450 1.8019 1.2470 -1.2470 -1.8019 0.4450 2 0.4450 -1.8019 -1.2470 1.2470 1.8019 -0.4450 -2 -0.4450 1.8019 1.2470 -1.2470 -1.8019 0.4450 -2 -0.4450 1.8019 1.2470 -1.2470 -1.8019 0.4450 2 0.4450 -1.8019 -1.2470 1.2470 1.8019 -0.4450 -2 -0.4450 1.8019 1.2470 -1.2470 -1.8019 0.4450 2 0.4450 -1.8019 -1.2470 1.2470 1.8019 -0.4450
E7u* 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0
E8u* 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 -2 0.4450 1.8019 -1.2470 -1.2470 1.8019 0.4450 -2 0.4450 1.8019 -1.2470 -1.2470 1.8019 0.4450 -2 0.4450 1.8019 -1.2470 -1.2470 1.8019 0.4450 -2 0.4450 1.8019 -1.2470 -1.2470 1.8019 0.4450
E9u* 2 -0.8678 -1.2470 1.9499 -0.4450 -1.5637 1.8019 0 -1.8019 1.5637 0.4450 -1.9499 1.2470 0.8678 -2 0.8678 1.2470 -1.9499 0.4450 1.5637 -1.8019 0 1.8019 -1.5637 -0.4450 1.9499 -1.2470 -0.8678 -2 0.8678 1.2470 -1.9499 0.4450 1.5637 -1.8019 0 1.8019 -1.5637 -0.4450 1.9499 -1.2470 -0.8678 2 -0.8678 -1.2470 1.9499 -0.4450 -1.5637 1.8019 0 -1.8019 1.5637 0.4450 -1.9499 1.2470 0.8678
E10u* 2 -1.2470 -0.4450 1.8019 -1.8019 0.4450 1.2470 -2 1.2470 0.4450 -1.8019 1.8019 -0.4450 -1.2470 2 -1.2470 -0.4450 1.8019 -1.8019 0.4450 1.2470 -2 1.2470 0.4450 -1.8019 1.8019 -0.4450 -1.2470 -2 1.2470 0.4450 -1.8019 1.8019 -0.4450 -1.2470 2 -1.2470 -0.4450 1.8019 -1.8019 0.4450 1.2470 -2 1.2470 0.4450 -1.8019 1.8019 -0.4450 -1.2470 2 -1.2470 -0.4450 1.8019 -1.8019 0.4450 1.2470
E11u* 2 -1.5637 0.4450 0.8678 -1.8019 1.9499 -1.2470 0 1.2470 -1.9499 1.8019 -0.8678 -0.4450 1.5637 -2 1.5637 -0.4450 -0.8678 1.8019 -1.9499 1.2470 0 -1.2470 1.9499 -1.8019 0.8678 0.4450 -1.5637 -2 1.5637 -0.4450 -0.8678 1.8019 -1.9499 1.2470 0 -1.2470 1.9499 -1.8019 0.8678 0.4450 -1.5637 2 -1.5637 0.4450 0.8678 -1.8019 1.9499 -1.2470 0 1.2470 -1.9499 1.8019 -0.8678 -0.4450 1.5637
E12u* 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 -2 1.8019 -1.2470 0.4450 0.4450 -1.2470 1.8019 -2 1.8019 -1.2470 0.4450 0.4450 -1.2470 1.8019 -2 1.8019 -1.2470 0.4450 0.4450 -1.2470 1.8019 -2 1.8019 -1.2470 0.4450 0.4450 -1.2470 1.8019
E13u* 2 -1.9499 1.8019 -1.5637 1.2470 -0.8678 0.4450 0 -0.4450 0.8678 -1.2470 1.5637 -1.8019 1.9499 -2 1.9499 -1.8019 1.5637 -1.2470 0.8678 -0.4450 0 0.4450 -0.8678 1.2470 -1.5637 1.8019 -1.9499 -2 1.9499 -1.8019 1.5637 -1.2470 0.8678 -0.4450 0 0.4450 -0.8678 1.2470 -1.5637 1.8019 -1.9499 2 -1.9499 1.8019 -1.5637 1.2470 -0.8678 0.4450 0 -0.4450 0.8678 -1.2470 1.5637 -1.8019 1.9499
Math View
Phys View A
Phys View B
✅
Chem View A
Chem View B


Additional information

Number of symmetry elements h = 56
Number of classes, irreps n = 56
Number of real-valued irreducible representations n = 30
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E C28 C14 (C28)3 C7 (C28)5 (C14)3 C4 (C7)2 (C28)9 (C14)5 (C28)11 (C7)3 (C28)13 C2 (C28)15 (C7)4 (C28)17 (C14)9 (C28)19 (C7)5 (C4)3 (C14)11 (C28)23 (C7)6 (C28)25 (C14)13 (C28)27 i (S28)15 (S7)11 (S28)17 (S14)9 (S28)19 (S7)5 (S4)3 (S14)11 (S28)23 (S7)13 (S28)25 (S14)13 (S28)27 σh S28 S14 (S28)3 S7 (S28)5 (S14)3 S4 (S7)9 (S28)9 (S14)5 (S28)11 (S7)3 (S28)13



Genrate representation from irreducible representations


Ag Bg E1g* E2g* E3g* E4g* E5g* E6g* E7g* E8g* E9g* E10g* E11g* E12g* E13g* Au Bu E1u* E2u* E3u* E4u* E5u* E6u* E7u* E8u* E9u* E10u* E11u* E12u* E13u*



Direct products of irreducible representations




Error: Reduction failed

Reduction to integer numbers for irreducible representations failed.
Error Code: 1

Binary products
⊙ Ag Bg E1g* E2g* E3g* E4g* E5g* E6g* E7g* E8g* E9g* E10g* E11g* E12g* E13g* Au Bu E1u* E2u* E3u* E4u* E5u* E6u* E7u* E8u* E9u* E10u* E11u* E12u* E13u*
Ag
Characters of reducible representation
E C28 C14 (C28)3 C7 (C28)5 (C14)3 C4 (C7)2 (C28)9 (C14)5 (C28)11 (C7)3 (C28)13 C2 (C28)15 (C7)4 (C28)17 (C14)9 (C28)19 (C7)5 (C4)3 (C14)11 (C28)23 (C7)6 (C28)25 (C14)13 (C28)27 i (S28)15 (S7)11 (S28)17 (S14)9 (S28)19 (S7)5 (S4)3 (S14)11 (S28)23 (S7)13 (S28)25 (S14)13 (S28)27 σh S28 S14 (S28)3 S7 (S28)5 (S14)3 S4 (S7)9 (S28)9 (S14)5 (S28)11 (S7)3 (S28)13
6 -28.223 -12.995 -0.783 -0.783 -12.995 -28.223 49 -28.223 -12.995 -0.783 -0.783 -12.995 -28.223 49 19.411 22.477 0.112 0.112 22.477 19.411 49 19.411 22.477 0.112 0.112 22.477 19.411 25 41.034 7.513 5.453 5.453 7.513 41.034 25 41.034 7.513 5.453 5.453 7.513 41.034 25 41.034 7.513 5.453 5.453 7.513 41.034 25 41.034 7.513 5.453 5.453 7.513 41.034